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Research Article | Open Access
Ozan Özkan, Abdullah Selçuk Kurbanli, "On a System of Difference Equations", Discrete Dynamics in Nature and Society, vol. 2013, Article ID 970316, 7 pages, 2013. https://doi.org/10.1155/2013/970316
On a System of Difference Equations
Abstract
We have investigated the periodical solutions of the system of rational difference equations , and where .
1. Introduction
Recently, a great interest has arisen on studying difference equation systems. One of the reasons for that is the necessity for some techniques which can be used in investigating equations which originate in mathematical models to describe real-life situations such as population biology, economics, probability theory, genetics, and psychology. There are many papers related to the difference equations system.
In [1], Kurbanli et al. studied the periodicity of solutions of the system of rational difference equations
In [2], Çinar studied the solutions of the systems of difference equations
In [3, 4], Özban studied the positive solutions of the system of rational difference equations
In [5–16], Elsayed studied a variety of systems of rational difference equations; for more, see references.
In this paper, we have investigated the periodical solutions of the system of difference equations where the initial conditions are arbitrary real numbers.
2. Main Results
Theorem 1. Let , , , , , , , , and be arbitrary real numbers, and let be a solution of the system Also, assume that , , , and . Then, all six-period solutions of (5) are as follows:
Proof. For , we have For , assume that are true. Also, we have
Theorem 2. Let , , , , , , , , and be arbitrary real numbers, and let be a solution of the system Also, assume that , , , and . Then, all six-period solutions of (10) are as follows:
Proof. For , we have For , assume that are true. Also, we have
The following corollary follows from Theorem 1.
Corollary 3. The following conclusions are valid for :(i),(ii),(iii),(iv).
The following corollary follows from Theorem 2.
Corollary 4. The following conclusions are valid for :(i),(ii),(iii),(iv).
References
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Copyright
Copyright © 2013 Ozan Özkan and Abdullah Selçuk Kurbanli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
